Proportion technique is a realm of “harmonic technique” that appears
from the outside to be all numbers and formulae. In spite of this, it
is one of the most lively domains, if one starts by understanding the
concept of proportion = relationship in terms of its actual nature: an
exceptional multitude of possible forms based on principles that are
very simple because they are understandable, provable by anyone with
basic knowledge, and not very mathematical. In this movement back and
forth between numbers, the numbers change into distances and then into
tones; thus every individual number, distance, or tone must always
somehow agree with two, three, or more others, or else persuade these
to agree with it. This process has a seductive charm for those who
calculate, draw, and listen along with it; the reader should work over
the examples once again and experience them. In this way, he will
surely grasp their meaning himself.

In §24a, we discussed proportions while considering “harmonic
division-ratios” in the harmonic pencil of rays of the equal-tone
lines. Unfortunately, the study of proportion is taught perfunctorily
nowadays; textbooks usually devote only a few pages to it, and one wise
writer thought that 3-4 pages “might be further abbreviated without
repercussions” (Tropfke: Geschichte der Elementarmathematik, 1902, vol. I, p. 232). We shall see about that as we orient ourselves.

The three most important proportions, probably originating from Babylon, were known in the school of Pythagoras:

the arithmetic a - b = c - d

the geometric a : b = c : d

and the harmonic a : c = (a - b) : (b - c)

This looks very dry. What do these formulae say? First, we will try to clarify them with lines.

The Arithmetic Proportion

For the arithmetic proportion, we must draw two lines from which we can
derive another two; the remainder must be equal. For example:

Figure 176

If I subtract b (= 3) from a (= 4), the result is 1. Subtracting d (= 5) from c (= 6) also yields 1. This is exactly what the formula a - b = c - d
indicates. Here we see that it is not the segments or their ratios that
are equal, but instead their differences: that is the so-called
“arithmetic” proportion. Naturally, one can perform this subtraction
with any given numbers and segments; the only requirement is that the
same difference results on both sides of the equation. Surplus or
shortage, then, is the characteristic of this proportion, depending on
whether (in the above example) I judge by the two upper or two lower
lines. “Arithmetic” proportion is thus named because an arithmetic
series, for example the simple whole-number series 1 2 3 4 5 ...,
represents a so-called “continuous” arithmetic proportion, in which the
differences between the adjacent elements are always the same (in this
case, 1). The constituent “arithmetic mean” of such a series is

n =

a + b

2

i.e. each element of the series is half the sum of its neighbors, e.g.

4 =

3 + 5

2

Now, if we examine our diagram, we immediately notice that in the
frequency diagrams, all horizontal rows of the overtone type clearly
form continuous arithmetic proportions. This means that every overtone,
in terms of its frequency, is half the sum of its two neighboring
tones. In the string-length diagram, the “arithmetic” element is in the
vertical series.

The Geometric Proportion

Whereas the arithmetic proportion only has a “ratio” in the equality of
differences, the geometric proportion has, so to speak, direct ratios
in the segments themselves, which must also be equal. If a : b = c : d, then at the simplest we are looking for numbers (in this case also fractions) with the same value, e.g. 4/8 = 2/4, 3/6 = 6/12, 20/4 = 5/1,
etc. In purely numeric terms, they can even be checked in this way,
since the inner and outer elements of this proportion can be multiplied:

4 : 8 = 2 : 4 8 · 2 = 4 · 4 16 = 16 etc.

But we can go further, using a geometric proportion to exchange the
elements of one string with those of another, without changing their
content:

8 : 4 = 4 : 2 4 · 4 = 8 · 2 16 = 16

The inner value remains the same. Now when we look back to our diagram,
we will notice that all the rows standing perpendicular to the
generator-tone axis (in the frequency diagrams; for string length,
reciprocal numbers or tone-values) form continuous geometric
proportions, for example:

3/1 : 1/1 : 2/3

or

15/4 : 1/1 : 4/15

or

10/5 : 1/1 : 5/10

g : c : f

h′′ : c : des,,

c′ : c : c,

fifth fifth

semitone semitone

octave octave

Figure 177

which can, of course, also be written in the form 3/2g : 1/1c = 1/1c : 2/3f,
etc. As one can see, the interval is always the same for geometric
proportions in harmonic terms. The characteristic thing about this
proportion is not, as with the arithmetic proportion, the equality of
the surplus or shortage (the difference), but instead the equality of
the value of both in terms of set distances (numbers, tones). Today
this geometric proportion, in the form a : b = c : d, is used as the prototype of proportion itself: “a is to b as c to d,” in which one calls the two quotients a : b and c : d
a “ratio” and the equalization of two ratios a “proportion.” Every
arithmetic and algebra textbook covers the manipulations of which the
individual elements of “proportion” are capable, especially
interchangeability and the ability of the inner and outer elements to
be multiplied, as already mentioned. The concept of a “continuous” or
“constant” proportion exists, then, whenever the median elements are
equal, as in the above examples. Thus the “median” geometric
proportional of 2 numbers emerges. When a : x = x : b, then x2 = ab, or

x = √ab (geometric mean).

The median geometric proportion between 2 numbers is the square root of
the product of these numbers, which can be demonstrated with our
diagram. If a = 3/2 and b = 2/3, then x2 = 3/2 · 2/3 = 1, thus x = 3/2 · 2/3
= √1 = 1. In the diagram, of course, we find other geometric
proportions if we merely look for the ratios of equal intervals-for
example: 4/2c : 4/3f = 2/4c, : 3/4g, where the equality-ratio consists of fifths.

The Harmonic Proportion

The predominantly geometric laws of the third of the typical
proportions, known as the harmonic, were already discussed amply in
§24a. Its algebraic expression a : c = (a - b) : (b - c) means that in a three-element harmonic proportion (a, b, c)
the differences between the first and middle elements have the same
relationship to the difference between the middle and third as the
first element has to the third. If, in a three-element (continuous)
harmonic proportion, (a - x) : (x - b) = a : b, then by multiplication of the outer and inner elements, we find that (a - x)b = (x - b)a, thus ab - bx = ax - ab, so 2ab= ax + bx, therefore

x =

2 a b

a + b

(harmonic mean).

Look
once again, more closely, at the visual geometric derivation in §24a.
If we look for harmonic proportion in our frequency diagram, we find it
in all the vertical columns of the model of the undertone series. Each
aliquot series is a continuous harmonic proportion series. The
progression 1/11/21/3, brought to a common denominator, is 6/63/62/6. (6/6 - 3/6) : (3/6 - 2/6) = 6/6 : 2/6, therefore 3/6 : 1/6 = 6/6 : 2/6,thus the left and right sides become 3/6, fulfilling the harmonic proportion. In a similar way, with the introduction of tone-values, the progression 1/2c,1/3f,,1/4c,,, set to a common denominator 6/124/123/12, yields the harmonic proportion (6/12c, - 4/12f,,) : (4/12f,, - 3/12c,,) = 6/12c, : 3/12c,,, so 2/12f,, : 1/12f,,,, on right and left. Visually, it can be clarified thus:

Figure 178

Here,
as one can see, in contrast with arithmetic proportions, it is not the
differences but the ratios of the differences that are equal to the
ratios of the two pairs of section differences.

Figure 179

It is now important and interesting to investigate the relationships of
the three proportion types to one another. Viewed harmonically, Fig.
179 makes this relationship self-explanatory. In the P-diagram one can
draw horizontal, vertical, and perpendicular lines through any point on
the generator-tone line to find three intersecting “continuous”
arithmetic, harmonic, and geometric proportions in each case-for
example, the series through the point 8/8c, as illustrated in Fig. 179.

The great polarity of major and minor also once again emerges here: all
major series (in terms of frequency, as in our diagram) have the
arithmetic proportion as a constituent element in the background, and
all minor series have the harmonic, whereas the geometric proportion
invades them both from one direction and overshoots in the other,
holding together the coordinate structure.

For further investigations of the relationships of the three proportion
types, their median proportionals are most useful. We find a remarkable
conjunction in the circle (Fig. 180); here we also find the solution to
the problem of the geometric construction of these three “median”
proportionals.

The Median Proportionals

Figure 180

In the circle (Fig. 180), choose an arbitrary point V on the radius MW. With the same radius r,
draw an arc from V that intersects the circumference at A, and continue
the line AV in the opposite direction to point H. Finally, draw a
perpendicular line from V to G. In every case, VA will be the
arithmetic proportional, VG the geometric proportional, and VH the
harmonic proportional to the segments a and b marked off by point V on the diameter UW. Proof: VA is the arithmetic median because VA = the radius r. a + b = 2r, so

r = VA =

a + b

2

The
arithmetic mean of the reciprocal values of two quantities is equal to
the reciprocal value of the harmonic mean of both quantities, i.e.:

_1_

a

+

_1_

b

=

a + b

-----------

2

-------

2ab

Figure 181

∆VAU is similar to ∆VWH, since UAH = UWH and AUW = AHW (angles on the circumference on equal chords), therefore:

VA : VU = VW : VH or

r : a = b : VH

VH =

a b

r

; r =

a + b

2

, therefore

VH =

a b

a + b

2

VH =

2 a b

a + b

Thus VH must be the harmonic median. VG, the geometric median, results
from the right triangle UGW, whose height GV, according to a well-known
law of plane geometry, is equal to the square root of the product of
the sections that it marks, a and b.

Since, in Fig. 180, point V can be arbitrarily chosen, it is clear that
only certain numbers are suited for an “emmelic” ratio of all three
median proportionals. What was said in §24a.1 regarding the musical
special case of “harmonic proportion” also applies here. We will soon
see how the three “medians” relate to one another algebraically and
harmonically.

From examining Fig. 180, it is directly evident that the arithmetic
mean must always be greater than the geometric, and the geometric
greater than the harmonic:

a + b

2

> √ab >

_2ab_

a + b

Figure 182

and that the geometric mean is also the geometric mean of the arithmetic and harmonic means:

a + b

2

: √ab = √ab :

_2ab_

a + b

Figure 183

The correctness of this proportion comes from the equality of the
product of the inner and outer elements. We also come upon this
remarkable symmetry for the median proportions, which we have already
come to know from the continuous proportions belonging to them in the
partial-tone diagram. This indication of symmetry led A. von Thimus to
further investigate the behavior of the three proportion types in
algebraic abstract terms. In vol. II, p. 38 of his Harmonikale Symbolik,
Thimus writes: “To gain a deeper insight into the number-harmonic
construction of the musical system, we wish to illustrate the process
of the Medietäten
(= median proportionals) and third proportionals more generally, by
algebraic methods. […] For this we start with the most general ideas.
We place the measure of the width of the still undefined interval in
the middle 'as beginning of everything,' take the same constructive
primal tone of the one, α (alpha) and the other, ω (omega) from the
pole (̕αρχαί= beginning of the up and down way ̔οδòς̕άνω
κάτω) ... the geometric median will equal the square root √αω of the
product of the two outer terms. The arithmetic median is known to be
half the sum of the outer term, so

α + ω

2

(

=

α1 + ω1

α0 + ω0

)

Figure 184

As a general formula for the harmonic, the expression

2αω

α + ω

(

= α ω

α0 + ω0

α1 + ω1

)

Figure 185

is found.”

Thimus now proceeds as follows: first he writes the three median
proportionals between the two “poles” α and ω, adding the arithmetic in
parentheses as above, and replacing

α + ω

2

with

α1 + ω1

α0 + ω0

Figure 186

for reasons of symmetry. He notes the harmonic proportional replacement of

2 α ω

α + ω

with

α0 ω0

α1 ω1

α

. . . . . . . . . . .

√αω

. . . . . . . . . . . . . . .

ω 1st geometric medians

α

. . . . . . . . . . . . . . . . . . . . .

α1 + ω1

α0 + ω0

. . .

ω 1st arithmetic medians

α

. . .

α0 ω0

α1 ω1

. . . . . . . . . . . . . . . . . . . . . . .

ω 1st harmonic medians

Figure 187

Now he begins to proportionate according to the series. First he seeks the third arithmetic proportional x as the “second arithmetic median proportional” to

2 α ω

α + ω

and

α +ω

2

Figure 188

According to the law of arithmetic proportion, the median element

α + ω

2

(

=

α1 + ω1

α0 + ω0

)

Figure 189

must equal the sum of the first element

2α + ω

α + ω

(

= α ω

α0 + ω0

α1 + ω1

)

Figure 190

and the desired third element, divided by 2:

α = ω

2

=

2 α ω

α + ω

+ x

2

Figure 191

and therefore

x = 2

(

α + ω

2

)

-

2 α ω

α + ω

which yields:

(α + ω)2 - 2 α ω

α + ω

=

α2 + ω2

α + ω

=

α2 + ω2

α1 + ω1

Figure 192

If we seek the “second harmonic median proportional” y according to the equation

α : y =

α2 + ω2

α1 + ω1

: ω

Figure 193

then after calculation, we find:

y =

α ω

α1 + ω1

α2 + ω2

Figure 194

In a similar way, we find the third arithmetic median proportional:

α3 + ω3

α2 + ω2

Figure 195

the third harmonic median proportional:

α ω

α2 + ω2

α3 + ω3

Figure 196

and so forth. The final collective result of this proportioning of the formula is:

Figure 197

The brackets above the formula represent geometric proportions, those
further above represent arithmetic proportions, and those below
represent harmonic proportions. The collective formula obeys the law of
the hyperbola, and shows the inner symmetry of the three interrelated
proportion types most beautifully. If one substitutes α = 1 and ω = 2
in the formula, that being the framework-interval of the octave, the
result is the series:

1 ... 10/9 (5/9) 6/5 (3/5) 4/3 √2 3/25/39/5 ... 2

Figure 198

If we now pursue the two halves of this series to the right and left of
√2 in the “P” system, we will see that these two halves, 3/25/39/5 ... and 3/33/55/9 ... each lie upon a straight line going out from 1/1-a
very remarkable condition if we seek the possible “location” of √2, and
even more significant if we examine the two end-values which these
lines approach, or at which they also finish in the formula, namely α =
1 and ω = 2, as “limits” of these lines. The mathematically adept
reader may investigate whether this is true!

We can now operate a three-element continuous proportion with the third
proportionals instead of with the median proportionals, to obtain new
steps and values. For this proportion technique, which is often
important for practical harmonic tasks, one should note the following
scheme.

a) If I am seeking, for example, the third arithmetic proportionals downwards for the partial-tone coordinates 10/3a and 8/3f,
then because the denominators must be equal for a continuous arithmetic
proportion, and the numerators must be equidistant, I write:

6/3c ← 8/3f → 10/3a

Thus 6/3c is the desired third arithmetic proportional:

(10/- 8/- 6/).

b) Conversely, for a continuous harmonic proportion, the number-values
must be equal and the denominators must be equidistant. For example, if
I am looking for the third harmonic proportionals upwards from 9/10b, and 9/9c, I write:

9/10b, → 9/9c → 9/8d

9/8d is the desired harmonic proportional:

(/10 - /9 - /8).

c) If I seek the third geometric proportional, e.g. for 10 e and 12 g downwards, then I write:

x : 10e = 10e : 12g

x = 100/12 = 50/6 = 25/3cis

Thus 25/3cis
is the desired third geometric proportional. In pure tones (without
numbers) these are equal intervals, as noted above: i.e. the interval g-e is equal to the interval e-cis (minor
thirds). Without arithmetic calculation, then, I could continue the
example, saying that the further geometric proportionals to 10 e and 25/3cis must be ais, since cis-ais is another minor third; and so on.

If we once again use the two poles α and ω as starting points, setting α = 1 c and ω = 2 c, then we find the corresponding third proportionals by proportioning back and forth, as follows:

If one uses octave reduction to bring all the tones in this series into
the same octave, in their correct scale-arrangement, then the result is
the diatonic scale:

bˇ c d es f g a (bˇ),

which to our perception is a B-major (g-minor) scale for a generator-tone of 1/1c. For the ancient Greeks, however, it was a “Dorian” scale, whose characteristic is that with c as the generator-tone, it begins a whole-tone lower and proceeds like a B-major scale, or with d as the generator-tone, it begins with c and proceeds like C-major.
This, of course, only explains it in terms of today's very primitive
perception of scales. For the ancient Greeks, and for those nowadays
who truly understand “Gregorian chant,” these “church modes” are
completely independent sound-forms with different psychical
characteristics.

Thimus then continues by differentiating the above proportioning
further, and thus comes to a new elicitation of the so-called
“enharmonics” of ancient Greek tone-systems that was lost, or at least
no longer used, by the later Greeks (Aristoxenos, Ptolemy, etc.): an
exquisitely differentiated ratio-construction with deep speculative and
symbolic significance.

The reader who, armed with a beginner's background in algebra,
calculates the above formulae, works through the proportions, and
devotes himself to further related exercises, will certainly appreciate
the deeper understanding of the laws of harmonic series-construction
thus made possible, despite the apparent schoolbook-like circumstances.
In a textbook, these things cannot be ignored. And those for whom such
numbers and formulae are an expression of actual relationships, rather
than just something external, will share the author's astonishment and
wonder at the phenomenon of proportion itself, which is revealed in its
innermost nature in the above examples. Especially in the last formula,
the inner laws are perceived with redoubled effect. All other readers,
though, who follow the suggestion in the preface and skip over the
above pages, will be rewarded by the following ektypic excursions.

§28a. Ektypics. Plato

In his later work Timaeus,
Plato ascribes to Timaeus the following words on proportion: “two
things cannot be rightly put together without a third; there must be
some bond of union between them. And the fairest bond is that which
makes the most complete fusion of itself and the things which it
combines; and proportion is best adapted to effect such a union.”* And here a definition of geometric
proportion is implied: “For whenever in any three numbers, whether cube
or square, there is a mean, which is to the last term what the first
term is to it; and again, when the mean is to the first term as the
last term is to the mean-then the mean becoming first and last, and the
first and last both becoming means, they will all of them of necessity
come to be the same, and having become the same with one another will
be all one.”** (Example: 2/3f : 1/1c : 3/2g, and 1/1c : 2/3f = 3/2g : 1/1).
The fact that Plato described the unity of the geometric proportions
playing around the generator-tone line as the “fairest bond” is in
itself a proof for the “harmonics” of late Platonic thought, a proof
that will be further solidified in the harmonic analysis of the scale
of the Timaeus (§39).

It is not claiming too much to refer to the phenomenon of ancient Greek
thought and perception as a thought and perception in proportions, an
ascendance of the spiritual (logos)
to a harmonically conditioned interplay of sensuality, in which this
proportional relationship (in Greek, in fact, the word for proportion
is λόγος = Logos!) plays the role of psychically and spiritually saturated tectonics.

The actual theory of proportion was created by the Pythagoreans. Thimus
found the fundamentals for his formulae described above mainly in
Iamblichus's commentary on the arithmetic of Nicomachus, and Iamblichus
remarked explicitly that the three proportions originated from Babylon.
Cantor (I, 167) agrees with this, since arithmetic and geometric series
were already known in Babylon and Egypt. For us, however, it is
fundamental that the ancients' entire study of proportion was almost
always illustrated with the introduction of tone-values; indeed, it was
through this that the musical spirit and perception of the Greeks were
first made understandable. At the center of this harmonic concept of
proportion stood the so-called “harmonia perfecta maxima,” the
“tetraktys”:

6 : 8 = 9 : 12

which was revered as something holy. This tetraktys consists of the three proportions:

6 : 9 : 12 arithmetic proportion

f bˇ f

6 : 8 : 12 harmonic proportion

f c f

6 : 8 = 9 : 12 geometric proportion

f c bˇ f

(tones by string-length)

It produces the most important intervallic building blocks: the octave, fifth, fourth, and whole-tone.

The Greeks reached interesting conclusions from the discovery (or
rediscovery) of the concept of proportion, only a few of which we can
discuss here.

The Irrational

Firstly, the concept of the irrational numbers, which indeed belong in our continuous geometric proportion a : √ab : b,
in which the median element is usually “irrational,” i.e. inexpressible
in rational numbers. For this reason, this proportion is known as the
“geometric,” since it can only be understood and conceived
geometrically. The classical example for the irrational is the diagonal
of the square:

Figure 205

which, according to the Pythagorean theorem, yields the value √2, if the square's sides are all 1. E. Frank writes in his book Plato und die sogenannte Pythagoreer
(1923, p. 224): “The deep impression that this discovery [the
irrational numbers] made upon their time can still be seen in Plato's
writings. It pointed to the existence of spatial quantities that could
in no way be expressed with rational numbers. This discovery
necessitated a complete revolution of mathematical thought, if the
theorems of mathematics were to be absolutely universal, i.e. should
apply to both rational and irrational numbers. To satisfy this
requirement, the Pythagoreans created the new method of plane geometry
and the new theory of proportions.”

The new method of plane construction provided universal size-ratios, which today we notate algebraically-for example, x = √a or (a + b)2 = a2 + 2ab + b2-or
geometrically, as in the side of a square or with the so-called
“gnomon”; it was a type of geometric algebra. By these methods, the
irrational quantities could be handled geometrically, and we find them
perfected in Euclid's Elements.
The theory of proportions follows directly from this attempt to conquer
the irrational numbers. The irrational number √2 is the “median
geometric proportional” between 1 and 2. E. Frank (op. cit. 226) writes
further: “The irrational quantity √2 can be brought into an exact
mathematic relationship with the whole numbers 1 and 2 by means of this
so-called geometric proportion. This proportion (in general: a : b = b : c or a : b = c : d)
is however called “geometric” because the median proportional (√2) can
only be found in it through geometric construction; it cannot be found
or represented arithmetically through any kind of rational number, nor
harmonically as the string-length of a harmonic tone. Through the
concept of proportion, then, the relationship of quantities is made
understandable to us irrespective of whether they are rational or
irrational. On the basis of this (geometric) concept of proportion, the
Pythagoreans then reconstructed all of mathematics. This study of
proportion was primarily created by Hippasus and Archytas, and brought
to conclusion by Archytas's student Euxodus.”

Despite indications in the sources, E. Frank made the old mistake of
viewing only the harmonic proportion as musically derivable. The
ancients analyzed both the arithmetic and the geometric proportions
harmonically, as we saw above; the latter, in fact, does not always
need an irrational median or medians. In spite of this, it is indeed
the irrational quantity √αω that, to a certain extent, controls and
holds together the three harmonic proportions as beginning, end, and
middle, and as a secret generating “ineffable” center.

This conclusion, first made possible by harmonic analysis, contradicts
the usual opinion of historians wishing to find, in the discovery of
irrational numbers, an upheaval of the ancient worldview based only on
rational numbers. The exact opposite is the case: the harmonic
inclusion of the irrational numbers in the “cosmos” of the being-values
arranged between the two infinities (1/∞-1/1-∞/1)
strengthens ancient wisdom and perception in the presence of a
universal harmony, precisely in that regulating power that grants the
irrational numbers their value-emphasized place in the order of things.
Of course, the discovery and teaching of proportional methods also had
its practical significance. “For the Greeks, the inestimable value of
proportions lay in the fact that they presented, with their
metamorphoses, a substitute for our equations. This important
application explains the extensive treatment given to proportions by
the Greek mathematician Euclid (ca. 300 BC), and also by Arabic and
Medieval scholars. For a long time in the Middle Ages, and almost up to
modern times, when letter notation was finally incorporated into
equations, people continued to write the results in the form of
proportions, which took the place of our modern closed formulae. Only
in modern times has the space allotted to them been severely reduced”
(Tropfke: Geschichte der Elementar-Mathematik, 1902, I, p. 232).

“The segmentation of important problems of fractional calculation through the so easily aesthetically elicited proportion-harmonia, logos!-works in a similar way; for the Greeks, 4/5
was not a fraction, but the abstract relationship between two relative
quantities. We know from Euclid that the Greeks indicated numbers with
proportions of segments, but precisely because of this, the pure
relation, the indifference to absolute size, must have been a
particularly lively problem for them. Once again one can say: in all
this lie the presuppositions for Plato's study of the large-small, the hyle
[matter] of the extensive, that must occur to the Logos formerly
indifferent to size in order for itself to 'come into being,' hence
giving it reality and determination in the full sense. The necessary
meeting of these two principles-a grasp of the relationship of logos to physis
that is actually transcendent, i.e. based on the absolutely necessary
mutual relationship of both-remains the core problem of philosophy, for
Aristotle as well as for Plato.”

Stenzel
writes further in his magnificent work (p. 93): “The manifest
sensualization of the spiritual is the theme of all late Platonic
philosophy.” Furthermore (p. 125):

“The
late Platonic theories do not reveal the pallor of thought; on the
contrary: they are the strongest confirmation of Fichte's thesis that
the Greeks achieved an exquisite refinement of sensuality much earlier
than that of abstract thought. One can say further that the value of
their philosophy for all times, especially for today, lies in the fact
they were spared from the unavoidable fate of all intellectuality, the
atrophy of the sense of sight, or worse, substituting it with
unexplained individual intention, because this prominence of sensuality
finally led to a symmetry of all spiritual energies, whose purest
expression will always remain Plato's and Aristotle's.”

From our harmonic viewpoint, we can only agree with these penetrating
views. Unfortunately, Stenzel, like all previous philologists, knew
only of the periphery of the harmonic backgrounds of Greek thought, and
not of the center. The reader who has worked through harmonic
fundamentals based on Thimus's studies and this textbook will find
hardly anything more depressing than having to look over and work
through the forest of often primitive commentary, laboring with
insufficient acoustic tools in “dark” numeric-harmonic areas,
especially on the fragments of Philolaus, the Pythagoreans in general,
and the late philosophy of Plato. If one places all these ideas of the
golden age of Greek philosophy upon the background of the concept of
akróasis and its norms, its numerically precisely understandable and
psychically perceivable laws-a procedure that is not at all arbitrary,
but actually encouraged by hundreds of ancient sources-then this
“refinement of sensuality,” which Stenzel very rightly detected in
those ideas, will become graspable and understandable from a fully new,
central position, inwardly adequate to this thought and accessible in
the deeper sense to heart and mind. One then sees that Plato did not
“come upon the unhappy [!] idea, in his old age, of developing the
world of ideas as a number-system” (W. Windelband: Platon,
1905, p. 93). On the contrary, for Plato, number-harmonics was the bond
joining idea and sensuality, and it was Windelband himself who came
upon a very “unhappy idea.” In my essay on Pythagoras (in Abhandlungen),
I attempted for the first time to rectify the most important
Pythagorean theorems of akróasis, which indeed place the eye and ear on
an equal level with thought. But a harmonic analysis of all Greek
philosophy up to around Proclus would require half a life's work; this
analysis requires a precise familiarity with the fundamentals of
harmonics and
an equally great knowledge of philosophical and natural history,
supported by thorough language abilities. I am sure that whoever
undertakes this project will achieve results that place all of the
ancients' scientific, artistic, and philosophical thoughts and
experiences upon a new, central foundation, not as seen by our modern
thought but self-radiating from the cosmos of Greek thought.

After this digression, we now return to our proportions and will note a
few more typical examples for the ektypics of harmonic proportioning.

Cube

The twelve sides, eight corners, and six faces of the cube form a
harmonic proportion. Nicomachus and Iamblichus believed that on this
basis, the ancients spoke of a ̔αρμωνίαγεωμετρική
(geometric harmony); because the octave (12 : 6 = 2 : 1), the fifth (12
: 8 = 3 : 2), and the fourth (8 : 6 = 4 : 3) are all present in the
cube's ratios. As we saw in §24, geometry, through the significance of
harmonic proportion, is anchored in harmonics especially on its
projective side. Clavius, in his commentary on Euclid, concentrates
completely on a musical proportion, which emerges in a triangle M1 M2 M3
that bisects any triangle A B C and the corresponding segments on the
connecting lines of the corner points and the bisection points (Fig.
206).

Figure 206

Draw a triangle A B C of any size, then draw lines from the corners to the midpoints of the sides M1, M2, and M3. Join M1, M2, and M3
together to form a smaller triangle, and indicate the intersection
points with the first set of lines with HJK. Then set a compass, for
example, to OH, and one will see that if OH is the unit 1, HA is 3 and
OM is 2. The analogy also applies for the units OJ and OK. Here are the
proportion numbers:

OH = 1 c

OM1 = 2 c,

HA and HM1 = 3 f,,

OA = 4 c,,

M1A = 6 f,,,

These sectors of the middle line AM1, which also apply for the two others BM2 and CM3, set in relation to one another as string lengths, yield the ratios:

1/1c

1/2c′

1/3g′

1/4c′′

.

1/6g′′

2/1c,

2/2c

2/3g

2/4c′

.

2/6g′

3/1f,,

3/2f,

3/3c

3/4f

.

3/6c′

4/1c,,

4/2c,

4/3g,

4/4c

.

4/6g

.

.

.

.

.

.

6/1f,,,

6/2f,,

6/3c,

6/4f,

.

6/6c

Figure 207

Here I have written all possible proportions between one another, so
that his example shows how the material of our partial-tone
coordinates, albeit lacking thirds here, emerges from a very simple
notation of various proportional possibilities, with an elementary
geometric theorem well known to the ancients.

Tonally, the two fifths (inverted: fourths) emerge with their octaves
from this triangle theorem, beside the primal tone and its octave, as
well as the large whole-tone from the comparison of g to f (f to g). [Note: the reader who owns Hörende Mensch should be aware of an error on p. 121, line 7 from the bottom, corrected thus: a : (a + b) : (a + b + c).]

Since it was traditional for the Greeks to test all segments on the
monochord and to set the tone-values obtained in relation to one
another, we will set the segment-numbers and their squares together
with the corresponding tone-values (since for a right triangle, c2 = a2 + b2).

3 f,,

4 c,,

5 as,,,,

9 bˇ,,,,

16 c,,,,

25 fes,,,,,

Figure 209

We will then place these values, reduced by octaves (without octave signatures) analogous to the above, in the relationship

9/3f

9/4bˇ

9/5dˇ

16/3g

16/4c

16/5e

25/3cesˇ

25/4fes

25/5as

* 3/4f

3/5a

* 4/3g

* 4/5e

5/3es

* 5/4as

* 3/9g

* 3/16f

3/25cis

4/9d

* 4/16c

4/25gis

5/9b

* 5/16as

* 5/25e

* 9/16bˇ

9/25fis

* 16/9d

* 16/25gis

25/9ges

* 25/16fes

Figure 210

If we now eliminate the duplicates (*) and arrange the remaining
tone-values in scales, the result is a nicely differentiated chromatic
scale:

Figure 211

One can naturally explain this result by means of the permutation
possibilities of the triadic ratios 2, 3, and 5, and say that any
triangle would produce the same result with triadic ratio sides reduced
by octaves. However, we are morphologists, and we think in terms of the
form. If this scale emerges within a typical limit such as that of the
Pythagorean triangle 3 : 4 : 5, that is reason enough for anyone who
has even a minimal understanding of morphological laws to pay close
attention. This is an optically viewable figure of a highly
characteristic quality, namely a triangle, which satisfies the
Pythagorean theorem with the smallest whole numbers possible; thus
behind it lies an exceptionally normative configuration. The fact that
calculating this yields a chromatic scale-admittedly not tempered, but
with the finest enharmonic variants-should be food for thought for
those harmonic hobbyists who tinker so long with some arbitrary numbers
that a chromatic scale arises from them like a phoenix from the ashes,
or who manage without a normative proof from the outset, which the
cautious among them still do today, precisely because they do not know
of such a proof. (Note: in my Hörende Mensch,
p. 116-117 frequencies are used as a basis instead of string-length for
analysis of the Pythagorean triangle; the result is naturally the same
as here.)

Newton

Here we offer the reader an excerpt from Thimus (Harmonikale Symbolik I, 48ff.) showing that the great Newton himself was interested in number-harmonic problems. This is what Thimus wrote:

“The
discoverer of the law of gravity directed his attention especially to
the musical number-ratios of ancient times, and did not fail to connect
his speculations on the relationships between the laws of art and the
mathematical laws of nature with the observation of laws of musical
harmony after the manner of the ancients. A young scholar, John Harington,
who occupied himself with subject matter relevant to this, had devised
a compilation of those harmonic numbers which, on the basis of the 1st
book of the Elements of Euclid, theorem 47, could be found in the right
triangle from the measure of the sides of the square of the hypotenuse
and the other two sides, if the three sides of the triangle have the
ratio 3, 4, and 5. The ratios found in this manner through
geometric-linear construction were as completely developed and as
musically rich as those that furnished the geometric figure of the
so-called Helikon contrived by Ptolemy. Ptolemy left out the thirds and
sixths. The right-angle triangle also provided the numbers at the basis
of these intervals in the linear measure. Harington, in his letters to Newton,
had mentioned how strange it would be in principle that the ancients
should have remained unaware of the ratios of the two thirds and
sixths, in other words the arithmetic and harmonic divisions of the
interval of a fifth, in which the pivotal point for the modern
tone-system lies [we could substitute any naturally developed
tone-system]. Here he also mentioned mystical Biblical numbers and
emphasized how the ratios for the triple octave of the major thirds
(300 : 30 = 10 : 1) and for the major sixth (50 : 30 = 5 : 3)-the
inversion of the minor third-could be found in the numbers described in
Genesis as the measures of the Ark: 300 · 50 and 30, expressed as
string lengths. He asked Newton's opinion of this, and also stated the
view that in architecture the art-forms that were pleasing to our
senses might well have their deeper reasons in a definite agreement of
the chosen measure with the harmonic numbers of musical intervals.
Newton's reply to the young scholar's communication can be found in
Hawkins: General History of the Science and Practice of Music
(London 1776, vol. III, pp. 142-143). It was only because Newton was
too busy with his multifarious other occupations that he did not pursue
the subject of the communication and the cue therein and did not
produce a more detailed work. He said that earlier he had occupied
himself with investigations of a similar type, and would like to have
the leisure in future to apply himself to the same. Meanwhile, he would
continue contemplating this subject-matter; for these were serious
investigations which showed how great is the simple beauty in all works
of the Creator. Moreover, pleasure in the arts requires the closest
possible approach to simple, concise forms, and this requirement
became, from his point of view, most fulfilled in the visual arts, the
more the chosen ratios approached the harmonic ratios. In architecture,
arc segments present a more or less pleasing image, according to
whether the emerging curves give the observer the idea of the law on
which they are based. He assumes that Harington would have consulted
Kepler's, Mersenne's, and the other authors' writings about the matter.
Newton goes on to say that what Harington said about the
inconceivability of the ancients' ignorance of the ratios of both
thirds is quite right. It would be very surprising that such highly
talented mathematicians should not have discovered in such number
observations that if the ratio of the fifth 3 : 2 does not permit
division in this form of its expression [through whole-number
interpolation], however it immediately becomes possible with the
exchange of the last term in the same ratio 6 : 4; now [through
interpolation of the five] the ratios yield both the major third 5 : 4
and the minor third 6 : 5, in which the diatonic system first reaches
completion, and without which the tone-system of the ancients must have
been very incomplete. It appears amazing that those who brought their
inquiries to a conclusion in such a minute way had not gone to work
more precisely in the investigation of the larger intervals, whose
divisions, just mentioned, are so fruitful for modern harmonics.
Furthermore, he would be inclined to accept that certain universal laws
laid down by the Creator would be authoritative for the pleasantness or
unpleasantness of all our various sense-impressions; at least such an
assumption would not in any way infringe upon the belief in the wisdom
or omnipotence of God, but would stand much more in unison with the
simplicity showing itself everywhere in the macrocosm. The letter is
dated from May 30, 1693 and closes with the assurance that all further
results of the investigations into the subject of the communication
would interest Newton to a high degree. Newton, in his optics, based
the exposition of his theory of the prism's graduated breaking up of
the various color-rays upon a comparison of the graduations with the
ratios of the Doric diatonic scale. This will be discussed later, and
will be the subject of detailed disquisitions with which we intend to
conclude our continuing investigations of harmonics and the study of
ancient number-theory in the second part.”

Unfortunately, the “second part” mentioned by Thimus, i.e. the third volume of his Harmonikale Symbolik,
was never published, and so far the search for the supposedly existing
manuscript has proved fruitless. The letter from Newton to Harington
speaks for itself. If such a great spirit was not afraid to take
number-harmonic investigations seriously, and indeed to direct his
special attention to them, then this is a challenge and incentive for
us today to give a new foundation to the whole field of harmonics and
to draw the corresponding conclusions.

Geometric Harmonics

Further on in the domain of elementary geometry, there is a wealth of
typical harmonic features, especially in the so-called “curious
properties of the triangle,” the sections determined by them, and so
forth. This is actually expressed in the word “chordal” (from ̔ηχορδή = the string), and “hypotenuse” (from ̔υποτείνω
= to span, namely a string). The beginnings of the function-concept and
the seedlings of analytical geometry are also based on the ancient
concept of proportion, whose innermost nature the ancients knew how to
derive harmonically and substantiate, as we have seen. The formula for
changing a rectangle into a square, which we write today as the formula
x2 = ab, was expressed by the Greeks in the proportion a : x = x : b, etc. There is more on this in the textbooks on the history of mathematics cited in the bibliography for this chapter.

The Imaginary Number i

It is highly interesting to note the sudden emergence of the imaginary number i when the two division ratios of a harmonically constructed segment are set equal.

Figure 212

As we know, in the harmonically divided segment, AB and PQ,

AP

PB

=

AQ

QB

and

PA

AQ

=

PB

BQ

Figure 213

i.e.
the segment AB is divided inwards and outwards in the same ratio by PQ,
likewise PQ by AB, but only each one in itself; the two ratios are not
equal. This yields the numeric proof

2/1 = 6/3 and 2/6 = 1/3

therefore 2 and 1/3

Figure 214

One can now calculate one ratio with the other. If one writes

x =

AP

PB

and y =

PA

AQ

Figure 215

and then continues this with BP,

y =

PA

BP

·

BP

AQ

= x ·

BP

AQ

=

_x_

AQ

(AQ - BQ - AP)

= x

(

1 -

BQ

AQ

-

AP

AQ

)

= x

(

1 -

1

x

- y

)

Figure 216

Therefore,

y = x - 1 - xy or y + xy = x - 1

from which it follows that

y =

x - 1

x + 1

or y + 1 = x - xy

and thus

x =

1 + y

1 - y

If we now set the two ratios equal, i.e. set x = y, then

x =

x - 1

x + 1

x2 = - 1

x = y = √-1 = i

Figure 217

From all these examples, and those in §24, it is undoubtedly evident
that seen historically, the fundamentals of the “new” projective
geometry, the germ of the function-concept, analytical geometry, the
concept of the infinite, the irrational, and the imaginary (the number i),
are all based on a common denominator of the universal concept of
proportion. The fact that the Greeks did not yet need our modern
formulae makes no difference regarding the actual knowledge that they
obviously possessed about these things. The common view of Greek
thought is that it was directed only towards the finite, the
“rational”; but for us, from the harmonic standpoint, the derivation
and basing of all these very “modern” concepts upon that of harmonic
proportion is of exceptional importance. This shows that the concept of
the ratio in itself, which indeed is most distinctly characterized in
the interval, has a constructive significance in the whole structure of
mathematics.

Proportion as Value-Form

If we grasp the nature of proportion in the value-formal sense, as I tried to do in my Grundriß
(p. 277ff. and 280ff.) under the value-forms of “ratio-equalization”
and “being-proportion,” this meaning becomes almost universal. The
reader should use his own knowledge and experience, and the parts of Grundriß
just mentioned, to make up for the lack of space to discuss further
ektypic examples here. The “golden section,” harmonically a third-sixth
problem, was specially discussed in my Harmonia Plantarum, p. 148ff. In §29.1 and §55.8 we will discuss this further.

From our human viewpoint, the concept of the “relationship” of humans
to one another is nothing other than proportions of psychic
individualities, whose “tones” must or should be tuned to one another
so as to somehow satisfy the proportional relationship, not in a
mathematical but in a value sense. Anyone who is comfortable in a
circle of friends knows well enough how someone who does not belong in
the circle “disturbs” it, which says nothing about the disturber as an
autonomous being-value, but simply affirms the fact that the previously
existing proportion is no longer in tune when a foreign element is
introduced. But a thought that is new, unfamiliar, objectionable, also
acts as a power in the previously existing spiritual proportion, and
must either adjust to the existing elements (if it is too weak) or
else, by the strength of its own value, rectify the other elements
until they enter into a new proportion with it.

J.J. Rousseau

J.J. Rousseau, in his Contrat Social,
Book 3, Ch. 1, imagines the relationship of ruler (president),
government, and people in the form of a geometric proportion. “One can
clarify the relationships set forth with the image of a continuous
proportion (a : b = b : c), the outer terms a and c representing the people as ruler and the people as a crowd of individuals, while the government is the geometric mean b.
The government receives orders from the ruler and gives them to the
people, and thus the state is in a correct balance. Equality must exist
between the product of the government's power times itself (b2) and the product of the ruler's power over the people times the power of the individual (a · b). (From a : b = b : c follows a · c = b2.)” (Quoted from A. Speiser: Klassische Stücke der Mathematik, Zürich, 1925, p. 105.)

The domains that emerge in a new light upon the background of harmonic
proportions will be discussed later. The first, “harmonics and
architecture,” will be the subject of the next chapter. The proportions
of the human form will be brought into connection with the
“sound-image” in §38, and proportional relationships in the planetary
system will appear in §41 with the discussion of the Timaeus scale; further harmonic theorems must be discussed beforehand.

§28b. Bibliography

A. von Thimus, Harmonikale Symbolik, preface x ff., I, 118ff., 196ff., II, 32ff. H. Kayser, Hörende Mensch, Ch. 2; Klang, 20, 21, 36, 37, 51-2, 142, 145; Grundriß,
103, 155, 287. On mathematical proportions, see the familiar textbooks.
For a new discussion of Greek thoughts and perceptions from the
standpoint of akróasis, indispensable initial literary tools are: 1) A.
von Thimus, Harmonikale Symbolik; 2) Diels' Fragmente der Vorsokratiker; 3) Grundriß, vol. I; 4) E. Frank, Plato und die sogenannte Pythagoreer, 1923; 5) J. Stenzel, Zahl und Gestalt bei Plato und Aristoteles, 1924; and 6) the author's fundamentals of new harmonics compiled in his works, especially his essay on Pythagoras in Abhandlungen. Aristides Quintilianus's De Musica,
fundamental for the entire “atmosphere” of Pythagorean musical
philosophy, has been excellently translated into German by R. Schäfke
(Berlin 1937). Regarding Frank, the first thing to observe is that the
main value of his book is in its wealth of dates and references, and in
the fundamentally correct assessment of harmonics in Greek thought. As
far as I can see, even specialists reject his quirk of proving that the
“so-called” (!) Pythagoreans were mythical. Stenzel's work is valuable
as an excellent collection of sources; it also does justice through
many subtle and deep thoughts to the assessment of the “harmonic” side
of Plato's later philosophy in its numeric-spiritual aspect. But
neither Frank nor Stenzel knew that this actually had a technical
harmonic background, though Thimus had brought it to light long before
them, and for these reasons they often labor vainly with certain things
that a harmonic analysis could make very simple. But precisely for this
reason, their books are more important for us harmonists; once one has
studied through them, one often has the feeling that many false
conclusions, incorrect interpretations, and misunderstandings in
certain places can be rectified by the single concept of tone-number,
not to mention the various harmonic theorems themselves, which often
shed a brilliant light in the darkest places.

But above all, the harmonically versed philologist should make a new
foray into ancient mathematics, metrics, music theory, and natural
philosophy. It seems to me that there is a wealth of harmonic relics
here, in manuscripts or in published but little-known authors, with
which history not yet known where to start, simply because no one has
been able to arrange them into any “valid” science-and thus no one has
been interested in them.